Grigore Ro?u

1
Complete Categorical Deductionfor Satisfaction
as Injectivity

• Grigore Ro?u
• Univeristy of Illinois at Urbana-Champaign

2
About This Work
3
Overview

• Equational logics
• Equations as epimorphisms
• Satisfaction as injectivity
• Category theory
• Factorization systems
• Injectivity, projectivity
• Finiteness
• Categorical equational deduction
• 4 categorical inference rules
• Soundness Completeness
• Conclusion and future work

4
Equational Logics (I)

• Simple and expressive
• Models algebras complete deduction
• Specify any computable domain
• Efficiently implementable (rewriting)
• OBJ, CafeOBJ, Maude, Elan, ASF/SDF
• Plethora of variants unsorted, many-sorted,
  order-sorted, partial
• Goal Uniform, categorical approach
• Axiomatizability, complete deduction,
  interpolation

5
Equational Logics (II)

• Complete Deduction
• Reflexivity, symmetry, transitivity, congruence,
  substitution
• Important results hold for sets of eqns
• Birkhoff axiomatizability
• Unconditional equations ? Varieties
• Conditional equations ? Quasi-varieties
• Craig interpolation

6
Equations as Epimorphisms (I)

• Sets of equations more convenient!
• ???? ? if ? where ?,? ? ???? ? ????
• Unconditional equations ?
• ? epimorphisms of free source
• equation ? epimorphism
• ???? ? an equation
• ? generates a congruence ? on ????
• ? generates an epimorphism e ???? ? ?
• epimorphism ? equation
• e ???? ? ? an epimorphism
• e generates an equation ???? Ker(e)

7
Equations as Epimorphisms (II)

• Conditional equations ? epimorphisms
• equation ? epimorphism
• ???? ? if ? a conditional equation
• ? generates a congruence ? on ????
• ? generates a congruence ? on ???? /?
• ? generates an epimorphism e (???? /?) ? ?
• epimorphism ? equation
• e ? ? ? an epimorphism
• ? the set U(?), the carrier of ?
• ? T(?) ? ? the counit of adjunction
• e generates the equation ???? Ker(??e) if Ker(?)

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Category TheoryInjectivity, Projectivity

• ? a class of morphisms in C
• object ? is ?-injective iff
• object P is ?-projective iff
• free projective in general

????
?
?
?g
?h
?
????
?
?
?g
?h
P
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Satisfaction as Injectivity (I)

• Unconditional equations ???? ? ?
• ? epimorphisms e T(X) ? ?

Injectivity
Satisfaction
? ???? ? iff forall h ? ? U(?) forall tt
in ? h?t? h?t?
?
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Satisfaction as Injectivity (II)

• Conditional equations ???? ? if ? ?
• ? epimorphisms e ???? /? ? ?

Injectivity
Satisfaction
e
? ???? ? if ? iff forall h ? ? U(?) such
that h?? ?, it follows that h???
????/?
?
?
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Category TheoryFactorization Systems

• ?E,M? factorization system for C iff
• E ? C subcategory of epimorphisms
• M ? C subcategory of monomorphisms
• E and M contain all isomorphisms
• each f in C can be uniquely ?E,M?-factored

?
e
m
isomorphic
f
?
?
e, e ? E
m
e
m, m ? M
?
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Categorical Equational Deduction (I)

• Category C with f.s. ?E,M? and enough projectives
• Equations (E for sets of equations)
• e ? ? ? in E
• unconditional iff ? is E-projective
• If ? not E-projective, then eA P? ? A is its
  condition
• Models objects of C
• Satisfaction
• M e
• E e iff (M E implies M e)

e
?
?
iff M is e-injective
M
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Categorical Equational Deduction (II)

• Given set E in E, our goal is to find a
  derivation system for arrows e ? ? ?
  s.t. E e iff E - e
• As expected, rules have the form

e1,
e2
en
,
,

e
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Four Categorical Inference Rules
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Categorical Inference Rules1. Identity

• Captures reflexivity

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Categorical Inference Rules2. Union

• e1 U e2 is the smallest epi whose kernel
  includes those of e1 and e2
• Captures
• symmetry, transitivity, congruence

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Categorical Inference Rules3. Restriction

• Using union, one may prove more than one needs
• Restriction allows one to restrict oneself to a
  subset of equations

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Categorical Inference Rules4. E-substitution
(old called E-pushout)

• Published in CSL01, but limited
• Captures substitution, but only for
  unconditional equations in E

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Categorical Inference Rules4. E-substitution

• Captures substitution properly, both for
  unconditional and for conditional equations
  in E

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Soundness Theorem

• Let E - e iff e is derivable from E with the
  four rules above
• Identity
• Union
• Restriction
• E-Substitution
• Soundness E - e implies E e

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Category TheoryFiniteness

• ?I,?? directed iff has finite upper bounds
• Directed colimit colimit of D ? ?I,?? ? C

C ? ?C? is finitely presentable iff Hom(C,_) C
? Set preserves dir. colimits
??i
Di
D
C
??
colim?D?
Captures finite sets as special case
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Completeness Theorem

• e A ? ? is
• finite iff finite object in A? E
• of finite condition iff eA P? ? A is finite
• Completeness If
• E has only equations of finite condition
• e is finite
• then E e implies E - e

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Related Work

• Goguen Meseguer
• Completeness of many-sorted EL
• Banaskewski Herrlich and
• Adamek Rosicky and
• Andreka, Nemeti Sain
• Equations as epics, injectivity
• Axiomatizability
• Diaconescu
• Category-based equational logic

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Conclusion and Future Work

• Categorical equational logic
• Equations are epis e ? ? ? in E
• Unconditional iff A is E-projective
• Sound complete categorical deduction for
  conditional equations!
• Dualize results to coalgebra?